Market Power and Welfare in Asymmetric Divisible Good Auctions

Transcription

1 Market Power and Welfare in Asymmetric Divisible Good Auctions Carolina Manzano Universitat Rovira i Virgili Xavier Vives IESE Business School October 016 Abstract We analyze a divisible good uniform-price auction that features two groups each with a finite number of identical bidders. Equilibrium is unique, and the relative market power of a group increases with the precision of its private information but declines with its transaction costs. In line with empirical evidence, we find that an increase in transaction costs and/or a decrease in the precision of a bidding group s information induces a strategic response from the other group, which thereafter attenuates its response to both private information and prices. A "stronger" bidding group -which has more precise private information, faces lower transaction costs, and is more oligopsonistic- has more market power and so will behave competitively only if it receives a higher per capita subsidy rate. When the strong group values the asset no less than the weak group, the expected deadweight loss increases with the quantity auctioned and also with the degree of payoff asymmetries. Market power and the deadweight loss may be negatively associated. KEYWORDS: demand/supply schedule competition, private information, liquidity auctions, Treasury auctions, electricity auctions JEL: D44, D8, G14, E58 For helpful comments we are grateful to Roberto Burguet, Vitali Gretschko, Jacub Kastl, Leslie Marx, Meg Meyer, Antonio Miralles, Stephen Morris, Andrea Prat, and Tomasz Sadzik as well as seminar participants at the BGSE Summer Forum, Columbia University, EARIE, ESSET, Federal Reserve Board, Jornadas de Economía Industrial, Princeton University, UPF, and Queen Mary Theory Workshop. We are also indebted to Jorge Paz for excellent research assistance. Corresponding author: carolina.manzano@urv.cat. Address: Departament d Economia i CREIP, Facultat d Economia i Empresa, Universitat Rovira i Virgili, Av. Universitat 1, 4304-Reus Spain. Tel: , Fax: Financial support from project ECO P is gratefully acknowledged. Financial support from the Spanish Ministry of Economy and Competitiveness Ref. ECO P and from the Generalitat de Catalunya, AGAUR grant 014 SGR 1496, is gratefully acknowledged.

2 1 Introduction Divisible good auctions are common in many markets, including government bonds, liquidity refinancing operations, electricity, and emission markets. 1 In those auctions, both market power and asymmetries among the participants are important; asymmetries can make market power relevant even in large markets. However, theoretical work in this area has been hampered by the diffi culties of dealing with bidders that are asymmetric, have market power, and are competing in terms of demand or supply schedules in the presence of private information. This paper helps to fill that research gap by analyzing asymmetric uniform-price auctions in which there are two groups of bidders. Our aims are to characterize the equilibrium, to perform comparative statics and welfare analysis from the standpoint of revenue and deadweight loss, and finally to derive implications for policy. Divisible good auctions are typically populated by heterogenous participants in a concentrated market, and often we can distinguish a core group of bidders together with a fringe. The former are strong in the sense that they have better information, endure lower transaction costs, and are more oligopolistic or oligopsonistic than members of the fringe. As examples we discuss Treasury and liquidity auctions in addition to wholesale electricity markets. Treasury auctions have bidders with significant market shares. That will be the case in most systems featuring a primary dealership, where participation is limited to a fixed number of bidders this occurs, for example, in 9 out of 39 countries surveyed by Arnone and Iden 003. A prime example are US Treasury auctions, which are uniform-price auctions. In these auctions, the top five bidders typically purchase close to half of US Treasury issues Malvey and Archibald Armantier and Sbaï 006 test for whether the bidders in French Treasury auctions are symmetric; these authors conclude that such auction participants can be divided into two distinct groups as a function of a their level of risk aversion and b the quality of their information about the value of the security to be sold. One small group consists of large financial institutions, which possess better information and are willing to take more risks. Kastl 011 also finds evidence of two distinct groups of bidders in uniform-price Czech Treasury auctions. Other papers that report asymmetries between bidders in Treasury auctions include, among others, Umlauf 1993 for Mexico, Bjonnes 001 for Norway, and Hortaçsu and McAdams 010 for Turkey. 1 See Lopomo et al. 011 for examples of such auctions. The relatively small number of primary dealers makes the US Treasury market imperfectly competitive Bikhchandani and Huang Uniform-price auctions are often used in Treasury, liquidity and electricity auctions, for example. See Brenner et al. 009 for Treasury auctions, with the United States a leading example since November Experimental work has found substantial demand reduction in uniform-price auctions see e.g. Kagel and Levin 001; Engelbrecht-Wiggans et al

3 Bindseil et al. 009 and Cassola et al. 013 find that the heterogeneity of bidders in liquidity auctions is relevant. Cassola et al. 013 analyze the evolution of bidding data from the European Central Bank s weekly refinancing operations before and during the early part of the financial crisis. The authors show that effects of the 007 subprime market crisis were heterogeneous among European banks, and they conclude that the significant shift in bidding behavior after 9 August 007 may reflect a change in the cost of short-term funding on the interbank market and/or a strategic response to other bidders. In particular, Cassola et al. 013 find that one third of bidders experienced no change in their costs of short-term funds from alternative sources; this means that their altered bidding behavior was mainly strategic: bids were increased as a best response to the higher bids of rivals. 3 Concentration is high also in other markets, such as wholesale electricity. This issue has attracted attention from academics and policy makers alike. A number of empirical studies have concluded that sellers have exercised significant market power in wholesale electricity markets see e.g. Green and Newbery 199; Wolfram 1998; Borenstein et al. 00; Joskow and Kahn Most wholesale electricity markets prefer using a uniform-price auction to using a pay-as-bid auction Cramton and Stoft 006, 007. In several of these markets e.g., California, Australia, generating companies bid to sell power and wholesale customers bid to buy power. In such markets, asymmetries are prevalent. For example, some generators of wholesale electricity rely heavily on nuclear technology, which has flat marginal costs, whereas others rely mostly on fuel technologies, which have steep marginal costs. Holmberg and Wolak 015 argue that, in wholesale electricity markets, information on suppliers production costs is asymmetric. For evidence on the effect of cost heterogeneity on bidding in wholesale electricity markets, see Crawford et al. 007 and Bustos-Salvagno 015. Our paper makes progress within the linear-gaussian family of models by incorporating bidders asymmetries with regard to payoffs and information. We model a uniform-price auction where asymmetric strategic bidders compete in terms of demand schedules for an inelastic supply we can easily accommodate supply schedule competition for an inelastic demand. Bidders may differ in their valuations, transaction costs, and/or the precision of their private information. 5 3 Bidder asymmetry has also been found in procurement markets, including school milk Porter and Zona 1999; Pesendorfer 000 and public works Bajari European Commission 007 has asserted that at the wholesale level, gas and electricity markets remain national in scope, and generally maintain the high level of concentration of the pre-liberalization period. This gives scope for exercising market power Inquiry pursuant to Article 17 of Regulation EC No 1/003 into the European gas and electricity sectors Final Report, Brussels, One reason for differences in private information among bidders may be the presence of both dealers and direct bidders in auctions such as in US Treasury auctions. Dealers aggregate the information of clients and bid with a higher precision of information for evidence from Canadian Treasury auctions, see Hortaçsu and

4 For simplicity and with an empirical basis, we reduce heterogeneity to two groups; within each group, agents are identical. We seek to identify the conditions under which there exists a linear equilibrium with symmetric treatment of agents in the same group i.e., we are looking for equilibria such that demand functions are both linear and identical among individuals of the same type. After showing that any such equilibrium must be unique, we derive comparative statics results. More specifically, our analysis establishes that the number of group members, the transactions costs, the extent to which bidders valuations are correlated, and the precision of private information affect the sensitivity of traders demands to private information and prices. When valuations are more correlated, traders react less to the private signal and to the price. We also find that the relative market power of a group increases with the precision of its private information and decreases with its transaction costs. Furthermore, if the transaction costs of a group increase, then the traders of the other group respond strategically by diminishing their reaction to private information and submitting steeper schedules. This result is consistent with the empirical findings of Cassola et al. 013 in European post-crisis liquidity auctions. If a group of traders is stronger in the sense described previously i.e., if its private information is more precise, its transaction costs are lower, and it is more oligopolistic, then the members of that group react more than do the bidders of the other group to the private signal and also to the price. This result may help explain the finding of Hortaçsu and Puller 008 for the Texas balancing market where, there is no accounting for private information on costs that, small firms use steeper schedules than the theory would predict. 6 We also find when the expected valuations between groups differ that the auction s expected revenue needs not be decreasing in the transaction costs of bidders, the noise in their signals, or the correlation of values. These findings contrast with the results obtained when groups are symmetric. We bound the expected revenue of the auction between the revenues of auctions involving extremal yet symmetric groups. In this paper we consider large markets and find that, if there is both a small and a large group of bidders, then the former oligopsonistic group has more market power and yet even the latter large group does not behave competitively since it retains some market power due Kastl 01; for a a theoretical model see Boyarchenko et al Linear supply function models have been used extensively for estimating market power in wholesale electricity auctions. Holmberg et al. 013 provide a foundation for the continuous approach as an approximation to the discrete supply bids in a spot market. In their experimental work, Brandts et al. 014 find that observed behavior is more consistent with a supply function model than with a discrete multi-unit auction model. Ciarreta and Espinosa 010 use Spanish data in finding more empirical support for the smooth supply model than the discrete-bid auction model. 3

5 to incomplete information. We also prove that the equilibrium under imperfect competition converges to a price-taking equilibrium in the limit as the number of traders of both groups becomes large. Finally, we provide a welfare analysis. Toward that end, we characterize the deadweight loss at the equilibrium and show how a subsidy scheme may induce an effi cient allocation. We find that if one group is stronger as previously defined, then it should garner a higher per capita subsidy rate; the reason is that traders in the stronger group will behave more strategically and so must be compensated more to become competitive. The paper also underscores how the bidder heterogeneity in terms of information, preferences, or group size documented in previous work may increase deadweight losses. In particular, when the strong group values the asset at least as much as the weak group, the deadweight loss increases with the quantity auctioned and also with the degree of payoff asymmetries. Our work is related to the literature on divisible good auctions. Results in symmetric pure common value models have been obtained by Wilson 1979, Back and Zender 1993, and Wang and Zender 00, among others. 7 Results in interdependent values models with symmetric bidders are obtained by Vives 011, 014 and Ausubel et al. 014, for example. 8 Vives 011, while focusing on the tractable family of linear-gaussian models, shows how increased correlation in traders valuations increases the market power of those traders. Bergemann et al. 015 generalize the information structure in Vives 011 while retaining the assumption of symmetry. Rostek and Weretka 01 partially relax that assumption and replace it with a weaker equicommonality assumption on the matrix correlation among the agents values. 9 Du and Zhu 015 consider a dynamic 7 Wilson 1979 compares a uniform-price auction for a divisible good with an auction in which the good is treated as an indivisible good; he finds that the price can be significantly lower if bidders are allowed to submit bid schedules rather than a single bid price. That work is extended by Back and Zender 1993, who compare a uniform-price auction with a discriminatory auction. These authors demonstrate the existence of equilibria in which the seller s revenue in a uniform-price auction can be much lower than the revenue obtained in a discriminatory auction. According to Wang and Zender 00, if supply is uncertain and bidders are risk averse, then there may exist equilibria of a uniform-price auction that yield higher expected revenue than that from a discriminatory auction. 8 Ausubel et al. 014 find that, in symmetric auctions with decreasing linear marginal utility, the seller s revenue is greater in a discriminatory auction than in a uniform-price auction. Pycia and Woodward 016 demonstrate that a discriminatory pay-as-bid auction is revenue-equivalent to the uniform-price auction provided that supply and reserve prices are set optimally. 9 This assumption states that the sum of correlations in each column of this matrix or, equivalently, in each row is the same and that the variances of all traders values are also the same. Unlike our model, Rostek and Weretka s 01 model maintains the symmetry assumption as regards transaction costs and the precision of private signals. The equilibrium they derive is therefore still symmetric because all traders use identical 4

6 auction model with ex post equilibria. For the case of complete information, progress has been made in divisible good auction models by characterizing linear supply function equilibria e.g., Klemperer and Meyer 1989; Akgün 004; Anderson and Hu 008. An exception that incorporates incomplete information is the paper by Kyle 1989, who considers a Gaussian model of a divisible good double auction in which some bidders are privately informed and others are uninformed. Sadzik and Andreyanov 016 study the design of robust exchange mechanisms in a two-type model similar to the one we present here. Despite the importance of bidder asymmetry, results in multi-unit auctions have been difficult to obtain. As a consequence, most papers that deal with this issue focus on auctions for a single item. In sealed-bid, first-price, single-unit auctions, an equilibrium exists under quite general conditions Lebrun 1996; Maskin and Riley 000a; Athey 001; Reny and Zamir 004. Uniqueness is explored in Lebrun 1999 and Maskin and Riley 003. Maskin and Riley 000b study asymmetric auctions, and Cantillon 008 shows that the seller s expected revenue declines as bidders become less symmetric. On the multi-unit auction front, progress in establishing the existence of monotone equilibria has been made by McAdams 003, 006; those papers address uniform-price auctions characterized by multi-unit demand, interdependent values and independent types. 10 Reny 011 stipulates more general existence conditions that allow for infinite-dimensional type and action spaces; these conditions apply to uniformprice, multi-unit auctions with weakly risk-averse bidders and interdependent values and where bids are restricted to a finite grid. The rest of our paper is organized as follows. Section outlines the model. Section 3 characterizes the equilibrium, analyzes its existence and uniqueness, and derives comparative statics results. We address large markets in Section 4 and develop the welfare analysis in Section 5. Section 6 concludes. Proofs are gathered in the Appendix. The model Traders, of whom there are a finite number, face an inelastic supply for a risky asset. Let Q denote the aggregate quantity supplied in the market. In this market there are buyers of two types: type 1 and type. Suppose that there are n i traders of type i, i = 1,. In that case, if the asset s price is p, then the profits of a representative type-i trader who buys x i units of the strategies. 10 McAdams 006 uses a discrete bid space and atomless types to show that, with risk neutral bidders, monotone equilibria exist. The demonstration is based on checking that the single-crossing condition used by Athey 001 for the single-object case extends to multi-unit auctions. 5

7 asset are given by π i = θ i p x i λ i x i /. So, for any trader of type i, the marginal benefit of buying x i units of the asset is θ i λ i x i, where θ i denotes the valuation of the asset and λ i > 0 reflects an adjustment for transaction costs or opportunity costs or a proxy for risk aversion. Traders maximize expected profits and submit demand schedules, after which the auctioneer selects a price that clears the market. The case of supply schedule competition for inelastic demand is easily accommodated by considering negative demands x i < 0 and a negative inelastic supply Q < 0. In this case, a producer of type i has a quadratic production cost θ i x i + λ i x i /. We assume that θ i is normally distributed with mean θ i and variance σ θ, i = 1,. The random variables θ 1 and θ may be correlated, with correlation coeffi cient ρ [0, 1]. Therefore, covθ 1, θ = ρσ θ.11 All type-i traders receive the same noisy signal s i = θ i + ε i, where ε i is normally distributed with null mean and variance σ ε i. Error terms in the signals are uncorrelated across groups covε 1, ε = 0 and are also uncorrelated with valuations of the asset covε i, θ j = 0, i, j = 1,. In our model, two traders of distinct types may differ in several respects: different willingness to possess the asset θ 1 θ, different transaction costs λ 1 λ, and/or different levels of precision of private information σ ε 1 σ ε. Applications of this model are Treasury auctions and liquidity auctions. For Treasury auctions, θ i is the private value of the securities to a bidder of type i; that value incorporates not only the resale value but also idiosyncratic elements as different liquidity needs of bidders in the two groups. For liquidity auctions, θ i is the price or interest rate that group i commands in the secondary interbank market which is over-the-counter. Here λ i reflects the structure of a counterparty s pool of collateral in a repo auction. A bidder bank prefers to offer illiquid collateral to the central bank in exchange for funds; as allotments increase, however, the bidder must offer more liquid types of collateral which have a higher opportunity cost. 3 Equilibrium Denote by X i the strategy of a type-i bidder, i = 1,, which is a mapping from the signal space to the space of demand functions. Thus, X i s i, is the demand function of a type-i bidder 11 The value of ρ will depend of the type of security. In this sense, Bindseil et al. 009 argue that the common value component is less important in a central bank repo auction than in a T-bill auction. 6

8 that corresponds to a given signal s i. Given her signal s i, each bidder in a Bayesian equilibrium chooses a demand function that maximizes her conditional profit while taking as given the other traders strategies. Our attention will be restricted to anonymous linear Bayesian equilibria in which strategies are linear and identical among traders of the same type for short, equilibria. Definition. An equilibrium is a linear Bayesian equilibrium such that the demand functions for traders of type i, i = 1,, are identical and equal to X i s i, p = b i + a i s i c i p, where b i, a i, and c i are constants. 3.1 Equilibrium characterization Consider a trader of type i. If rival s strategies are linear and identical among traders of the same type and if the market clears, that is, if n i 1X i s i, p + x i + n j X j s j, p = Q, for j = 1, and j i, then this trader faces the residual inverse supply p = I i + d i x i, where I i = n i 1 b i + a i s i + n j b j + a j s j Q / n i 1 c i + n j c j and d i = 1/ n i 1 c i + n j c j. The slope d i is an index of the trader s market power. 1 As a consequence, this trader s information set s i, p is informationally equivalent to s i, I i. The bidder therefore chooses x i to maximize E [π i s i, p] = E [θ i s i, I i ] I i d i x i x i λ i x i /. The first-order condition FOC is given by E [θ i s i, I i ] I i d i x i λ i x i = 0, or equivalently, X i s i, p = E [θ i s i, p] p / d i + λ i. 1 The second-order condition SOC that guarantees a maximum is d i + λ i > 0, which implies that d i + λ i > 0. Using the expression for I i and assuming that a j 0, we can show that s i, p is informationally equivalent to s 1, s. Therefore, since E [θ i s i, p] = E [θ i s i, I i ], it follows that According to Gaussian distribution theory, where Ξ i = with σ ε i = σ ε i /σ θ and σ ε j E [θ i s i, p] = E [θ i s 1, s ]. E [θ i s i, s j ] = θ i + Ξ i si θ i + Ψi sj θ j, 3 1 ρ + σ ε j and Ψ i = 1 + σ ε i 1 + σ ε j ρ 1 We assume that n i 1 c i + n j c j 0. ρ σ ε i, 1 + σ ε i 1 + σ ε j ρ = σ ε j /σ θ. We remark that Equation 3 has the following implications. 7

9 1. The private signal s i is useful for predicting θ i whenever 1 ρ + σ ε j 0, that is, when either the liquidation values are not perfectly correlated ρ 1 or type-j traders are imperfectly informed about θ j σ ε j 0.. The private signal s j is useful for predicting θ i whenever ρ σ ε i 0, that is, when the private liquidation values are correlated ρ 0 and type-i traders are imperfectly informed about θ i σ ε i 0. Our first proposition summarizes the previous results. It shows the relationship between a i and c i in equilibrium and also indicates that these coeffi cients are positive. Proposition 1. Let ρ < 1. In equilibrium, the demand function of a trader of type i i = 1,, X i s i, p = b i + a i s i c i p, is given by X i s i, p = E [θ i s i, p] p / d i + λ i, with d i +λ i > 0, d i = 1/ n i 1 c i + n j c j, and a i = i c i > 0, for i = 1 / ρ 1 σ ε i. The coeffi cient c i can be expressed as a function of the ratio z = c 1 /c, and z is the unique positive solution of a cubic polynomial p z = 0. The equilibrium demand function depends on E [θ i s i, p]. As for the price coeffi cient see Expression 11 in the Appendix, c i = 1 Ψ i n i c i + n j c j n j a j 1/ d i + λ i, note that the term Ψ i n i c i + n j c j n j a j 1 is the information-sensitivity weight of the price. Note also that, the more informative the price higher Ψ i n i c i + n j c j n j a j 1, the lower the price coeffi cient lower c i. Furthermore, this term vanishes when Ψ i = 0, that is, when either the valuations are uncorrelated ρ = 0 or the private signal s i is perfectly informative σ ε i = 0 since in those cases the price conveys no additional information to a trader of type i. Since a i > 0 and c i > 0, for i = 1,, it follows that in equilibrium the higher the value of the trader s observed private signal or the lower the price, the higher the quantity she will demand. When σ ε i > 0 we have a i < c i, since i < 1 in this case; when σ ε i = 0, we have i = 1 and a i = c i. Observe that we can write the demand as X i s i, p = b i + c i i s i p. Because p is a linear function of s 1 and s, for i = 1, we have E [θ i s i, p] = E [θ i s 1, s ] i.e., Equation holds. The equilibrium price is therefore privately revealing, in other words, the private signal and the price enable a type-i trader to learn as much as about θ i if she had access to all the information available in the market, s 1, s. If ρ = 0 or if both signals are perfectly informative σ ε i = 0, i = 1,, then bidders do not learn about θ i from prices. Hence, E [θ i s i ] = E [θ i s i, p] = E [θ i s 1, s ] for i = 1,. The demand functions are given by X i s i, p = E [θ i s i ] p / d i + λ i, i = 1,. 8

10 Hence, c i = 1/ d i + λ i and so, given our expression for d i, we have d i = 1 /n i 1 / d i + λ i + n j / d j + λ j for i, j = 1, and j i. We can show that, this system has a unique solution satisfying the inequality d i +λ i > 0, i = 1,, iff n 1 + n 3. In this case, the equilibrium coincides with the full-information equilibrium denoted by superscript f. 13 Furthermore, when ρ = 0, market power d i is independent of σ ε i, i = 1, ; and when σ ε i = 0, for i = 1,, market power is independent of ρ. Our next proposition describes the condition under which an equilibrium exists and shows that, if an equilibrium does exist, then it is unique. Proposition. There exists a unique equilibrium if and only if z N > z D, where z N and z D denote the highest root of, respectively, q N z and q D z, with q N z = n Ξ n Ξ1 1 1 n 1 1 n z n Ξ n1 z and q D z = n n 1 1 Ξ 1 + n1 Ξ 1 n 1 n + 1 z + n 1Ξ 1 z. Let z = c 1 /c. Then, in equilibrium, z D < z < z N, lim λ1 0 z = z N and lim λ 0 z = z D. For an equilibrium to exist we must have c i > 0 i = 1, and these inequalities hold if and only if iff z D < z < z N. No equilibrium exists for ρ close to 1 or for low n 1 + n. Neither does an equilibrium exist when ρ = 1. If the price reveals a suffi cient statistic for the common valuation, then no trader has an incentive to place any weight on her signal. But if traders put no weight on signals, then the price contains no information about the common valuation. This conundrum is related to the Grossman-Stiglitz 1980 paradox. Remark 1. If n 1 = 1 and n = 1, then z N = 1/ 1 Ξ and z D = Ξ 1 1. Since i Ξ 1 i > 1, i = 1,, we can use direct computation to obtain z N < z D. Applying Proposition, we conclude that no equilibrium exists in this case. Therefore, the inequality n 1 + n 3 is a necessary condition for the existence of an equilibrium in our model. This result is in line with Kyle 1989 and Vives To develop a better understanding of the equilibrium and the condition that guarantees its existence, we consider two particular cases of the model: a monopsony competing with a fringe; and symmetric groups. 13 In the full shared information setup, traders can access s 1, s. In this framework the price does not provide any useful information. 14 Du and Zhu 016 consider ex post nonlinear equilibria in a bilateral divisible double auction. 9

11 Monopsony competing with fringe Corollary 1. For n = 1 the equilibrium exists if 1 ρ > ρ 1 σ ε 1 and n 1 > n 1 ρ, σ ε 1, σ ε, where n1 increases with ρ, σ ε 1, and σ ε. If, also, λ = 0 and σ ε = 0, then n 1 ρ, σ ε 1, 0 / = 1 + ρ σ ε 1 1 ρ ρ 1 σ ε 1 and x = c θ p, with c = n 1 c 1. An equilibrium with linear demand functions exists provided there is a suffi ciently competitive trading environment n 1 high enough. In the particular case where λ = 0 and σ ε = 0, expressions for the equilibrium coeffi cients can be characterized explicitly see the Appendix. From the expressions for c i i = 1, it follows that, if n 1 = n 1, then the equilibrium cannot exist because in this case the demand functions would be completely inelastic c i = 0, i = 1,. Symmetric groups Consider the following symmetric case: n = n 1 = n, λ 1 = λ = λ, and σ ε 1 = σ ε = σ ε. Here z = 1 in equilibrium. From Proposition we know that, if an equilibrium exists, then the value of z is in the interval z D, z N. It follows that z N > 1 > z D or, equivalently, that q N 1 > 0 and q D 1 > 0. After performing some algebra, we find that the foregoing inequalities are / satisfied iff n > 1+ρ σ ε 1 ρ 1 + ρ + σ ε, where σ ε = σ ε/σ θ. Therefore, the equilibrium s existence is guaranteed provided either that n is high enough or that ρ or σ ε is low enough. Vives 011 also analyzes divisible good auctions with symmetric bidders, but in his model the bidders receive different private signals. The condition that guarantees existence of an / equilibrium in Vives setup is n > + M, where M = nρ σ ε 1 ρ 1 + n 1ρ + σ ε. Direct computation yields that the condition derived in the model of Vives is more stringent than the condition derived in our setup. The reason is that, in Vives 011, the degree of asymmetry in information and induced market power is greater because each of the n traders receives a private signal. The rest of this subsection is devoted to describing some properties that satisfy the equilibrium coeffi cients and then to comparing the equilibrium quantities. Comparative statics We start by considering how the model s underlying parameters affect the equilibrium and, in particular, market power Proposition 3. We then explore how the equilibrium is affected when there are two distinct groups of traders, that is, a strong group and a weak group Corollary. Proposition 3. Let ρσ ε 1 σ ε > 0. Then, for i = 1,, i j, the following statements hold. i An increase in θ i or Q, or a decrease in θ j, raises the demand intercept b i. 10

12 ii An increase in λ i, λ j, σ ε i, σ ε j, or ρ makes demand less responsive to private signals and prices lower a i and c i and increases market power d i. iii If σ ε i and/or λ i increase, then d i /d j decreases. iv If n i and/or n j increase, then d i decreases. Remark. If ρ = 0, then: a both c i and d i as well as c j and a j, j i are independent of σ ε i ; b a i decreases with σ ε i ; and c b i is independent of both Q and θ j. If σ ε i = 0 for i = 1,, then b i = 0, and c i, c j, a i, a j, d i, and d j, i = 1,, j i, are independent of ρ. That is, for the information parameters to matter for market power, it is necessary that prices convey information. And given that the equilibrium values of d 1 and d when ρ = 0 or when σ ε i = 0, i = 1, are equal to those corresponding to the full-information setup, Proposition 3ii implies that, if ρσ ε 1 σ ε > 0, then d f i < d i, i = 1,. Thus, in this case asymmetric information increases the market power of traders in both groups beyond the full-information level. By Lemma A1, the only equilibrium coeffi cient affected by the quantity offered in the auction Q and by the prior mean of the valuations θ i and θ j is the coeffi cient b i. Proposition 3i indicates that if Q increases, then all the bidders will increase their demand higher b 1 and b. Moreover, if the prior mean of the valuation of group i increases, then the bidders in this group will demand a greater quantity of the risky asset higher b i. Then the intercept of the inverse residual supply for the group j bidder rises in response to a higher θ i. That reaction leads the traders in group j to reduce their demand for the risky asset lower b j. Part ii of Proposition 3 shows how the response to private information and price varies with several parameters. If the transaction costs for a bidder increase, then that bidder is less interested in the risky asset and so a i and c i are each decreasing in λ i. Moreover, any increase in a group s transaction costs also affects the behavior of traders in the other group. If λ i increases, then c i decreases, in which case the slope of the inverse residual supply for group j increases higher d j. This change leads group-j traders to reduce their demand sensitivity to signals and prices lower a j and c j. We can therefore see how an increase in the transaction costs for group-i traders say, a deterioration of their collateral in liquidity auctions that raises λ i leads not only to steeper demands for bidders in group i but also, as a reaction, to steeper demands for group-j traders. Figure 1 illustrates the case of initially identical groups that become differentiated after a shock induces a higher λ 1 and also raises group s willingness to pay for liquidity as in a crisis situation both θ 1 and θ, which affect the intercepts of the demand functions. 11

13 Figure 1: Equilibrium demand functions for ρ = 0.75, σ θ = 5, Q = 4, n i = 5, σ ε i = 1, and s i = θ i, i = 1,. We also analyze how the response to private information and price varies with a change in the precision of private signals. If the private signal of type-i bidders is less precise higher σ ε i, then their demand is less sensitive to private information and prices. Thus a trader finds it optimal to rely less on her private information when her private signal is less precise. A private signal of reduced precision also gives the type-i bidder more incentive to consider prices when predicting θ i, which leads in turn to this bidder having a steeper demand function lower c i. The same can be said for a bidder of type j because of strategic complementarity in the slopes of demand functions. 15 We also find that the more highly the valuations are correlated higher ρ, the less is trader responsiveness to private signals lower a i, i = 1, and the steeper are inverse demand functions lower c i, i = 1,. We can explain these results by recalling that, when the valuations are correlated ρ > 0, a type-i trader learns about θ i from prices. In fact, the price is more informative about θ i when ρ is larger, in which case demand is less sensitive to private information. The rationale for the relationship between the correlation coeffi cient ρ and the slopes of demand functions is as follows. An increment in the price of the risky asset makes an agent more optimistic about her valuation, which leads to less of a reduction in demand quantity than 15 This result in the supply competition model may help explain why, in the Texas balancing market, small firms use steeper supply functions than predicted by theory Hortaçsu and Puller 008. Indeed, smaller firms may receive lower-quality signals owing to economies of scale in information gathering. 1

14 in the case of uncorrelated valuations. 16 Proposition 3iii states that any increase in the signal s noise or in group i s transaction costs has the effect of reducing its relative market power, since then the ratio d i /d j i j decreases. Finally, part iv formalizes the anticipated result that an increase in the number of auction participants higher n i or n j reduces the market power of traders in both groups. Corollary. Suppose that σ ε 1 σ ε, λ 1 λ, and n 1 n, and suppose that at least one of these inequalities is strict. Then, in equilibrium, the following statements hold. i The stronger group here, group reacts more both to private information and to prices a 1 < a, c 1 < c and has more market power d 1 < d than does the weaker group. ii The value of the difference d 1 + λ 1 d + λ is, in general, ambiguous. If 1 ρ n 1 n 1 + ρ + σ ε 1 n 1 ρ + σ ε 1 + n1 ρ σ + 1 ρ n 1 n ρ + σ ε ε 1 n 1 1 ρ + σ ε + n ρ σ 1, 4 ε then d 1 + λ 1 < d + λ always holds. Otherwise, d 1 + λ 1 > d + λ iff λ 1 /λ is high enough. Part i of this corollary shows that if a group of traders is less informed, has higher transaction costs, and is more numerous, then it reacts less both to private signals and to prices. Observe in particular that group-1 traders, having less precise private information, rely more on the price for information higher Ψ 1 n 1 c 1 + n c n a 1 ; as a result, their overall price response c 1 = 1 Ψ 1 n 1 c 1 + n c n a 1 / d 1 + λ 1 is smaller. Similarly, group-1 traders, for whom n 1 is larger, put more information weight on the price which depends more strongly on s 1. Corollary ii is useful for comparing allocations across groups. It indicates that the inequality d 1 + λ 1 > d + λ holds whenever a the differences between groups stem mainly from transaction costs; 17 and b λ 1 /λ is high enough. If signals are perfect σ ε i = 0, i = 1, or if ρ = 0, then part i holds and d 1 + λ 1 > d + λ iff λ 1 > λ. Equilibrium quantities Finally, we examine the equilibrium quantities. Let t i = E [θ i s 1, s ], i = 1,, be the predicted values with full information s 1, s. After some algebra, it follows that equilibrium quantities are functions of the vector of predicted values t = t 1, t : x i t = n j t i t j d j + λ j + n i d j + λ j + n j d i + λ i n }{{} i d j + λ j + n j d i + λ i Q, i = 1,, j i. 5 }{{} x I i t x C i t 16 A high price conveys the good news that the private signal received by other group s traders is high. When valuations are positively correlated, a bidder infers from the high private signal of the other group that her own valuation is high. 17 This claim follows because if n 1 = n and σ ε 1 = σ ε, then the inequality given in 4 does not hold. 13

15 Observe that, according to these expressions, the equilibrium quantities can be decomposed into two terms: a valuation trading term and a clearing trading term, which we denote by respectively x I i t and x C i t for group i, i = 1,. With regard to the information trading term, it vanishes when t 1 = t, but has a positive resp. negative value for the group with the higher resp. lower value of t i. Moreover, n 1 x I 1 t + n x I t = 0. As for the clearing trading term, we remark that it vanishes when Q = 0; otherwise, it is positive for both groups yet lower resp. higher for the group with higher resp. lower d i +λ i. In addition, n 1 x C 1 t+n x C t = Q. Taking expectations in Equation 5, we have E [x 1 t] E [x t] = n 1 + n d + λ d 1 + λ 1 θ1 θ + n 1 d + λ + n d 1 + λ 1 n 1 d + λ + n d 1 + λ 1 Q. Group 1 trades more when it values the asset more highly θ 1 > θ and when its traders are less cautious d + λ > d 1 + λ 1 than group. By combining Corollary with the equation just displayed, we obtain the following remarks. Remark 3. If Q is low enough, then E [x 1 t] > E [x t] whenever θ 1 > θ. In contrast, if Q is high enough, then E [x 1 t] > E [x t] whenever d +λ > d 1 +λ 1. Under the assumptions of Corollary, this latter inequality is satisfied if 4 holds or if λ 1 /λ is suffi ciently low. Remark 4. When Q = 0 i.e., the so-called double auction case, then E [x t] < 0 < E [x 1 t] iff θ 1 > θ. Then group 1 consists of buyers and group of sellers. 3. Bid shading, expected discount, and expected revenue Our aim here is to identify factors that affect the magnitudes of bid shading, expected discount, and expected revenue. Let t = n 1 t 1 + n t / n 1 + n. From the demand of bidders it follows that p t = t i d i + λ i x i t, i = 1,. Therefore, p t = t d 1 + λ 1 n 1 x 1 t + d + λ n x t /n 1 + n. 6 Bid shading For a trader of type i, the expected marginal benefit of buying x i units of the asset is t i λ i x i. Hence, the average marginal benefit is given by t λ 1 n 1 x 1 + λ n x / n 1 + n. The magnitude of bid shading is the difference between the average marginal valuation and the auction price, that is, d 1 n 1 x 1 + d n x / n 1 + n. We can use Equation 5 to write bid shading as n d d 1 + λ 1 + n 1 d 1 d + λ n 1 + n n 1 d + λ + n d 1 + λ 1 Q + t t 1 d d 1 n n 1 n 1 + n n 1 d + λ + n d 1 + λ 1. 7 At this juncture, some additional remarks are in order. 14

16 Bid shading increases with Q. When d 1 = d = d as in the symmetric case, for instance, bid shading consists of only one term the first one and it is equal to dq/ n 1 + n. When d 1 d, the second term of 7 is negative and bid shading decreases whenever the group that values the asset more highly t i > t j has less market power d i < d j. If group 1 has higher transaction costs λ 1 > λ, is more numerous n 1 > n, and is less informed σ ε 1 > σ ε than group, then c 1 < c, and so d 1 < d. If t 1 > t, then the second term of 7 is negative and the two terms have opposite signs. Therefore, if Q is low e.g., Q = 0 or if the difference in predicted values of the asset is high, then negative bid shading obtains. Expected discount The expected discount is defined as E [ t ] E [p t]. We can use Equation 6 to write the expected discount as d 1 + λ 1 n 1 E [x 1 t] + d + λ n E [x t] / n 1 + n. Now some algebra yields the following expression for the expected discount: d 1 + λ 1 d + λ n 1 d + λ + n d 1 + λ 1 Q + n 1n d + λ d 1 λ 1 θ θ 1 n 1 + n n 1 d + λ + n d 1 + λ 1. 8 Here our related comments are as follows. When d 1 + λ 1 = d + λ = d + λ as in the symmetric case, the expected discount is d + λq/ n 1 + n. The first term is always positive provided Q > 0, whereas the second term is positive whenever d + λ d 1 λ 1 θ θ 1 > 0. Therefore, the expected discount is lower whenever the group that values the asset more highly θ > θ 1 has a lower "total transaction cost" d + λ < d 1 + λ 1. If group 1 ex ante values the asset more θ 1 > θ, has higher transaction costs λ 1 > λ, is more numerous n 1 > n, and is less informed σ ε 1 > σ ε, then Corollary shows that d 1 + λ 1 > d + λ whenever a the differences between groups are due mostly to transaction costs and b λ 1 /λ is high enough. In this case, both terms are positive and so the expected discount is positive. Yet, if both groups have similar transactions costs, then the two terms in 8 have opposite signs. In particular, we expect a negative discount when Q is low. 15

17 Expected revenue The expected price is given by n1 E [p] = θ 1 + n / n1 θ Q + n. d 1 + λ 1 d + λ d 1 + λ 1 d + λ It is worth noting that, in the double auction case Q = 0, E [p] is a convex combination of θ 1 and θ. Also, for symmetric groups except possibly with respect to the means we have E [p] = θ 1 + θ /. The seller s expected revenue is E [p] Q and, provided that he has enough supplies at no cost, the revenue-maximizing supply is given by Q = 1 n 1 d 1 +λ 1 θ 1 + n d +λ θ. That supply Q is increasing in the expected valuations and in the number of traders; it is decreasing in those traders market power and transaction costs. Proposition 4. Let ρσ ε 1 σ ε > 0. Then, in equilibrium, the following statements hold. i If θ 1 = θ, then the expected price is increasing in n i but is decreasing in λ i, σ ε i, and ρ, i = 1,. Otherwise, if θ 1 θ is large enough, then these results need not hold. ii The expected revenue: - increases with θ i for i = 1,, and increases with Q for E [p] > 0; - is between a the larger expected revenue of the auction in which both groups are ex ante identical with a large number of bidders each group with max {n 1, n }, high expected valuation max { θ 1, θ }, low transaction costs min {λ1, λ } and precise signals min { σ ε 1, σ ε } and b the smaller expected revenue of the auction in which both groups are ex ante identical but with the opposite characteristics i.e., min {n 1, n }, min { θ 1, θ }, max {λ1, λ }, and max { σ ε 1, σ ε }. Remark 5. If ρ = 0, then E [p] is independent of σ ε i i = 1,, and if σ ε i = 0, i = 1,, then E [p] is independent of ρ. The reason is that in both cases, d i is independent of σ ε i and ρ. Proposition 4 indicates that the relationship between expected price on the one hand and λ i, σ ε i, and ρ, i = 1, on the other hand is potentially ambiguous. For example, if θ θ 1 is high enough, then E [p] is decreasing in n 1 ; yet, if θ 1 = θ, then the derived results are in line with those in the symmetric case, where E [p] = θ d + λ Q/n see Vives 010, Prop.. We should like to understand how ex ante differences among bidders affect the seller s expected revenue. Suppose that group is our strong group; it has lower transaction costs λ < λ 1, is less numerous n < n 1, and is better informed σ ε < σ ε 1. If this group values the asset less, θ < θ 1 resp., values it more, θ > θ 1, then expected revenue is lower resp., higher than in the case where θ 1 = θ. If θ 1 θ, then Proposition 4i suggests that group s relatively small size n < n 1 reduces the seller s expected revenue, although both its relatively low transaction costs λ < λ 1 and its relatively precise signals σ ε 16 < σ ε 1 have the opposite

18 effect. So in general, the ex ante differences between the two groups have an ambiguous effect on the seller s expected revenue. Nonetheless, part ii of Proposition 4 directly follows from part i. 4 Large markets Our objective in this section is to determine whether or not the equilibrium under imperfect competition converges to a price-taking equilibrium in the limit as the number of traders becomes large. We examine two possible scenarios: in the first, only group 1 is large; in the second, both groups of bidders are large. The per capita supply denoted by q is assumed to be inelastic, that is, Q = n 1 + n q. 4.1 Oligopsony with competitive fringe Proposition 5. Let ρσ ε 1 σ ε > 0. Suppose that n 1 and n <. Then an equilibrium exists iff n > n ρ, σ ε 1, σ ε, where n is increasing in ρ and σ ε 1 and where n is decreasing in σ ε whenever ρ 1 σ ε 1 < 1 ρ. An agent in the large group absorbs the inelastic per capita supply in the limit lim b 1 = q, lim a 1 = lim c 1 = 0 and retains some market power n 1 n 1 n 1 lim d 1 > 0, while an agent in the small group commands a higher degree of market power n 1 lim d > lim d 1. n 1 n 1 When n = 1, the existence condition stated in Proposition 5 boils down to ρ 1 σ ε 1 < 1 ρ from Corollary 1. Equation 3 shows that, when n = n ρ, σ ε 1, σ ε, the demand functions for bidders in group would be completely inelastic lim c = 0. This explains n 1 why the inequality n > n ρ, σ ε 1, σ ε is required for the existence of equilibrium. Neither group 1 nor group has flat aggregate demand in the limit, and each group has some market power. We see that an agent in the large group just absorbs the inelastic per capita supply, behaving like a "Cournot quantity setter", and keeping some market power lim d 1 > 0, while n 1 bidders in the small group command relatively more market power lim d > lim d 1. It is n 1 n 1 worth to remark that the large group retains market power in the limit only if there is learning from the price incomplete information and correlation of values, ρσ ε 1 σ ε > 0. In this case the aggregate demand of group 1 does not become flat, lim n 1c 1 <. Otherwise, lim n 1c 1 = n 1 n 1 and lim d 1 = 0. It is easy to see also that, in the limit, the price depends only on the valuations n 1 and market power of agents in the competitive fringe: lim p = E [θ 1 s 1, s ] lim d 1 + λ 1 q. n 1 n 1 17

19 If the small group is fully informed σ ε = 0 and the large group is entirely uninformed σ ε 1, then: n = ρ; an equilibrium always exists for n > ; and the equilibrium coeffi cients for group are lim b = 0, and lim a = lim c = n 1 n 1 n 1 groups relative market power is given by lim d /d 1 = 1 + ρ. n 1 n ρ 4. A large price-taking market n ρ n ρλ. In this case, the Consider now the following setup. There is a continuum of bidders along the interval [0, 1], and we let q denote the aggregate average quantity supplied in the market. Suppose that a fraction µ i 0 < µ i < 1 of these bidders are traders of type i, i = 1,. Then the following proposition characterizes the equilibrium of this continuum economy and shows that it is the limit of a finite economy s equilibrium. Proposition 6. Let Q = n 1 + n q. Suppose that n 1 and n both approach to infinity and that n i /n 1 + n converges to µ i 0 < µ i < 1 for i = 1,. Then, the equilibrium coeffi cients converge to the equilibrium coeffi cients of the equilibrium of the continuum economy setup, which are given by b i = c i = σ ε i ρλj q + µ j θi ρθ j, µ i ρλ j σ ε i + µ j λ i 1 ρ + σ a i = ε i µ j 1 ρ 1 + ρ + σ ε i µ j 1 ρ µ i ρλ j σ ε i + µ j λ i 1 ρ + σ ε i, and µ i ρλ j σ ε i + µ j λ i 1 ρ + σ ε i, where i, j = 1,, j i. 5 Welfare analysis This section focuses on the welfare loss at the equilibrium. We characterize the equilibrium and effi cient allocations in Subsection 5.1 and analyze deadweight losses in Subsection Characterizing the equilibrium and effi cient allocations Recall that t i = E [θ i s 1, s ], i = 1,, that is, the predicted values with full information s 1, s and t = t 1, t. The strategies in the equilibrium induce outcomes as functions of the realized vector of predicted values t and are given in Equation 5. One can easily show that the equilibrium outcome solves the following distorted benefit maximization program: 18 max x 1,x E [ n 1 θ1 x 1 d 1 + λ 1 x 1/ + n θ x d + λ x / t ] 18 See Lemma A3 in the Appendix. s.t. n 1 x 1 + n x = Q, 18

20 where d 1 and d are the equilibrium parameters. The effi cient allocation would obtain if we set d 1 = d = 0, which corresponds to a price-taking equilibrium denoted by superscript o. The equilibrium strategy of a type-i bidder i = 1, will be of the form X o i s i, p = b o i + a o i s 1 c o i p, i = 1,, and is derived by maximizing the following program: max x i E [θ i s i, p] p x i λ i x i /, while taking prices as given. The FOC of this optimization problem yields E [θ i s i, p] p λ i x i = 0. After identifying coeffi cients and solving the corresponding system of equations, we find that there exists a unique equilibrium in this setup. The equilibrium coeffi cients coincide with those in Proposition 6 for the continuum market. Proposition 7. Let Q = n 1 +n q and let µ i = n i /n 1 +n for i = 1,. Then there exists a unique price-taking equilibrium, and the equilibrium coeffi cients coincide with the equilibrium coeffi cients of the continuum setup whose expressions are given in the statement of Proposition 6. Our next corollary provides some comparative statics results. Corollary 3. Let ρσ ε 1 σ ε > 0. Then the only equilibrium coeffi cients affected by Q, θ i, and θ j are the intercepts of the demand functions with b o i increasing in θ i and Q and decreasing in θ j for i, j = 1, and i j. Furthermore, the demands of group i are less sensitive to private signals and prices lower a i and c i in response to an increase in λ i, λ j, ρ, σ ε i, and µ i, and to a decrease in µ j ; however, group i s demands are not affected by σ ε j. Observe that, under competitive behavior, we can derive an additional comparative statics result: the relationship between the equilibrium coeffi cients and the proportion of individuals in group 1. In particular, increasing the proportion µ 1 of type-1 traders leads, for those traders, to an increased information-sensitivity weight of the price higher Ψ 1 n 1 c o 1 + n c o n a o 1 and so a lower overall response to the price c o 1 = λ Ψ1 n 1 c o 1 + n c o n a o 1 ; the opposite holds for type- traders. Thus the auction outcome can be obtained as the solution to a maximization problem with a more concave objective function than the expected total surplus, which suggests that ineffi ciency may be eliminated by quadratic subsidies κ i x i /, i = 1, that compensate for the distortions. The per capita subsidy rate κ i to a trader of type i must be such that it compensates for the distortion d i κ i while accounting for the subsidy. Since the aim is to induce competitive behavior, the trader should be led to respond with c o i to the price. This means that the exact 19